polynomialmatricestest.nb

单模多项式矩阵的分解算法

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Cell["Some examples on how to use all these functions follow.", "Text",
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Cell[CellGroupData[{

Cell["Hermite, Smith and McMillan forms", "Section",
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Cell["\<\
These three functions have two versions: one computing only the \
canonical form\
\>", "Text",
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Cell[BoxData[
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Cell[BoxData[
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Cell["\<\
and an extended version returning the unimodular matrices that lead \
to that form too\
\>", "Text",
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Cell["{h,u}=ExtendedHermiteForm[m]", "Input",
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Cell[BoxData[
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Cell["h===Expand[u.m]", "Input",
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Cell["u is unimodular", "Text"],

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Cell["Det[u]", "Input",
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Cell["\<\

The function HermiteForm computes the Hermite form of matrices with dependent \
columns as well:\
\>", "Text",
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  Evaluatable->False],

Cell[CellGroupData[{

Cell[BoxData[
    \(HermiteForm[{{x, x\^2, x}, {x\^2, x\^3, x + 1}, {x\^3, x\^4, x + 2}}] // 
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Cell["\<\
The functions that compute the Smith and McMillan forms can be used \
the same way\
\>", "Text",
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Cell["\<\
SmithForm[m={{x-1,x+1,0},{x,x-2,1},{x+1,2x,1}}]//MatrixForm
MatrixForm/@({s,u,v}=ExtendedSmithForm[m])
s===Expand[u.m.v]\
\>", "Input",
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          RowSpacings->1,
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Cell[BoxData[{
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        m = DiagonalMatrix[{1 - x, 1 - x, 1 - x, 
              1 - x, \(-2\) - x, \(-2\) - x}]] // MatrixForm\), "\n", 
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          RowBox[{"(", "\[NoBreak]", GridBox[{
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                {\(-1\), 
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Cell[BoxData[
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2)\)\^2\)\)}}] // MatrixForm\)], "Input"],

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Cell["\<\
MatrixForm/@({mc,u,v}=ExtendedMcMillanForm[m])
mc===Simplify[u.m.v]\
\>", "Input",
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Cell["\<\
These functions work on matrices whose elemtents are polynomials \
with symbolic coefficients as well; in this case one must specify the \
indeterminate, otherwise an error message is displayed:\
\>", "Text",
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  Evaluatable->False],

Cell[CellGroupData[{

Cell[BoxData[
    \(SmithForm[
      m = {{x\^2 + d\ x + k, k, 0}, {d\ x, k, 1}, {x, x, d}}]\)], "Input"],

Cell[BoxData[
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      "The argument has more than one variable. One more specifying an \
indeterminate is expected."\)], "Message"],

Cell[BoxData[
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Cell["The right input is", "Text",
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Cell[CellGroupData[{

Cell["SmithForm[m,x]", "Input",
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Cell[BoxData[
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Cell[CellGroupData[{

Cell["Division", "Section",
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Cell["\<\
The result of LDivision[a,b] is a list {q,r} consisting of the \
quotient and the remainder obtained by left division of matrix a by matrix b \
(a=b.q+r). The matrices a and b must have the same number of rows.\
\>", \
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  Evaluatable->False],

Cell[CellGroupData[{

Cell[BoxData[{
    \({q, r} = 
      LDivision[a = {{x\^2 + 2  x\^4}, {1 + x + x\^2}}, 
        b = {{x\^2 + 1, 1}, {1, x\^2}}]\), "\[IndentingNewLine]", 
    \(a === Expand[b . q + r]\)}], "Input"],

Cell[BoxData[
    \({{{\(-1\) + 2\ x\^2}, {\(-1\)}}, {{2}, {2 + x}}}\)], "Output"],

Cell[BoxData[
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Cell["\<\
The functions LQuotient and LRemainder compute only the left \
quotient and remainder respectively:\
\>", "Text",
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  Evaluatable->False],

Cell[CellGroupData[{

Cell["\<\
LQuotient[a,b]
LRemainder[a,b]\
\>", "Input",
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Cell[BoxData[
    \({{\(-1\) + 2\ x\^2}, {\(-1\)}}\)], "Output"],

Cell[BoxData[
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Cell["\<\
LGCD computes the greatest common left divisor of two polynomial \
matrices\
\>", "Text",
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  Evaluatable->False],

Cell[CellGroupData[{

Cell[BoxData[{
    \(g = 
      LGCD[a = {{x\^2 + x}, {x\^2 + x}}, 
        b = {{x\^2, x}, {x\^2 - x + 1, x}}]\), "\[IndentingNewLine]", 
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Cell[BoxData[
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Cell[BoxData[
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Cell[BoxData[
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Cell["\<\
The least common left multiple of two polynomial matrices is given \
by Llcm\
\>", "Text",
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  Evaluatable->False],

Cell[CellGroupData[{

Cell[BoxData[
    \(Llcm[{{x - 1, 1}, {x + 1, 0}}, {{x\^2, 1}, {1, 0}}]\)], "Input"],

Cell[BoxData[
    \({{1 + x, 0}, {\(-1\) + x, 1}}\)], "Output"]
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Cell["\<\
The function LCoprime is a left coprimality test for polynomial \
matrices\
\>", "Text",
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  Evaluatable->False],

Cell[CellGroupData[{

Cell[BoxData[
    \(LCoprime[{{x, x + 1, x - 2}, {0, x, 0}}, {{x, x - 1}, {x\^2, 
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Cell[BoxData[
    \(True\)], "Output"]
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Cell["\<\
The functions for the right division are: RDivision, RQuotient, \
RRemainder, RGCD, Rlcm, RCoprime and their usage is the same.\